A meteorological model gives the air temperature T (in degrees Celsius) at a high-altitude research station as a function of time t (in hours past midnight). The model is piecewise-defined for one 24-hour period by
T(t)=⎩⎨⎧20+10sin(9πt),8+2t−18,0≤t≤18,18<t≤24.
Assume that temperature readings are recorded continuously whenever the function is defined.
(1) Part 1: (a) Verify analytically whether T is continuous at t=18.
(2) Part 2: (b) Determine all critical numbers of T in the open interval (0,24) and classify each as a relative maximum, relative minimum, or neither. Justify your classifications.
(3) Part 3: (c) Using all necessary evaluations, determine the highest temperature and the lowest temperature recorded during the 24-hour period, and specify the exact times at which they occur.
(4) Part 4: (d) The cold-weather equipment at the station automatically activates when the temperature first drops below 6∘C. Does the equipment activate during this 24-hour period? Support your conclusion with evidence from your work in part (c).